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Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.

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Free CAS Exam 5 Formula Sheet PDF, No Signup (2026)

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All Exam 5 Formulas

Ratemaking Data & Exposures 17 items

Calendar year incurred loss

L_{C Y} = Paid_{year} + (R_{end} - R_{begin})

, Paid = losses paid during the year,

R_{e n d}

= ending case reserve,

R_{b e g in}

= beginning case reserve

Customer lifetime value closed form

CLV = \frac{m}{1 - r v}

, m = constant annual margin per policy, r = retention rate, v = annual discount factor

Transition-year claims-made rate from the cumulative report-lag step factor

Rate_{n} = (\sum_{j = 0}^{n - 1} p_{j}) \times Mature Rate

p_{j}

= share of ultimate cost at report lag j, n = policy maturity year

Excess ratio for large losses

Excess Ratio = \frac{\sum losses above threshold}{\sum losses capped at threshold}

, threshold = large-loss cap; loaded losses = capped losses

\times (1 + Excess Ratio)

Additive expense fee (policy fee) under the variable permissible loss ratio

Fee = \frac{F}{1 - V - Q}

, F = average fixed expense per exposure, V = variable expense %, Q = target profit %, denominator = variable permissible loss ratio

Off-balance factor for class relativity changes

O B = \frac{1}{1 + Δ d}

, OB = off-balance factor applied to the base rate,

Δ d

= change in the exposure-weighted average rate differential

Classical partial credibility square-root rule

Z = \frac{n}{n _{full}}

, Z = credibility weight, n = observed claim count,

n_{full}

= full-credibility standard (e.g. 1,082 claims)

Exponentially trended value over a trend period

V_{future} = V_{base} (1 + r)^{t}

V_{ba se}

= base value, r = annual trend rate, t = trend period in years (midpoint to midpoint)

Parallelogram on-level factor

F = \frac{I _{c u r}}{I ˉ}

, F = on-level factor,

I_{c u r}

= current cumulative rate index,

I ˉ

= area-weighted average rate index for the experience period

Bühlmann credibility factor

Z = \frac{n}{n + K}, K = \frac{EPV}{VHM}

, n = claim count, K = Bühlmann constant, EPV = expected process variance, VHM = variance of hypothetical means

Frequency-severity pure premium cross check

\frac{C}{E} \times \frac{L}{C} = \frac{L}{E}

, C = claim count, E = earned exposures, L = incurred losses; frequency times severity must equal losses over exposures

Indicated rate change factor under the loss ratio method

CF = \frac{( L + E _{L} ) / P + F}{1 - V - Q}

, L = losses,

E_{L}

= LAE, P = on-level premium, F = fixed expense ratio, V = variable expense ratio, Q = profit provision; percent change = CF − 1

Actuarially sound rate satisfying Principle 2 (all costs)

R = \frac{P P + F}{1 - V - Q}

, PP = pure premium (loss+LAE), F = fixed expense per exposure, V = variable expense ratio, Q = profit and contingency provision

Exposure trend factor for an inflation-sensitive base

(1 + g)^{n}

, g = annual exposure inflation rate (wage or price growth), n = trend period in years from historical midpoint to future policy midpoint

Indicated rate under the pure premium method

R = \frac{P P + F}{1 - V - Q}

, R = indicated rate per exposure, PP = pure premium (losses+LAE per exposure), F = fixed expense per exposure, V = variable expense %, Q = profit and contingencies %

GLM factor relativity under a log link

R_{j} = e^{β_{j}}

R_{j}

= multiplicative relativity for the factor level,

β_{j}

= fitted log relativity (coefficient); base level has

β_{j} = 0

R_{j} = 1.00

Net cost of reinsurance load

N C = Ceded Prem - E [Recoveries]

, NC = net cost loaded per exposure, Ceded Prem = reinsurance premium ceded, E[Recoveries] = expected ceded recoveries

Topic 0 18 items

Expected emergence over an interim period

E_{emerge} = \frac{U - A _{t_{1}}}{1 - % _{t_{1}}} \times (%_{t_{2}} - %_{t_{1}})

, U = prior ultimate,

A_{t_{1}}

= actual emerged to date,

%_{t_{1}}

= percent emerged at last valuation,

%_{t_{2}}

= percent emerged at interim

Net unpaid claims from gross unpaid and unpaid recoveries

Net Unpaid = Gross Unpaid - Unpaid S&S - Unpaid Reins

, S&S = salvage & subrogation, Reins = reinsurance recoverable, all unpaid = ultimate minus received to date

On-level severity restated to a common cost level via an inflation index

S_{on-level} = S_{raw} \times \frac{I _{current}}{I _{claim year}}

S_{r a w}

= raw severity,

I_{c u r r e n t}

= index at target level,

I_{c l aim}

year = index at claim's cost level

Unpaid claim reserve from ultimate losses and percent paid

R = U \times (1 - p)

, R = unpaid claim reserve, U = ultimate losses, p = proportion of ultimate losses paid to date

Bornhuetter-Ferguson ultimate losses

U_{B F} = Reported + Premium \times ELR \times (1 - p)

, ELR = a priori expected loss ratio, p = 1/CDF = reported percentage, 1−p = unreported percentage

Unpaid ULAE under the classical paid-to-paid technique

Unpaid ULAE = R \times [0.50 \times C + 1.00 \times I]

, R = paid-ULAE-to-paid-loss ratio, C = case reserves, I = pure IBNR (claims not yet opened)

Total ultimate losses with a separate large-claim provision

U_{t o t a l} = L_{u l t} + P_{l a r g e}

U_{t o t a l}

= total ultimate,

L_{u l t}

= developed limited ultimate (limited losses × limited LDF),

P_{l a r g e}

= separately estimated large-claim provision

Berquist-Sherman adjusted reported losses for a cell

R_{a d j} = C ˉ_{a d j} (N - K_{a d j}) + P_{a d j}

C ˉ_{a d j}

= adjusted avg case,

N

= reported counts,

K_{a d j}

= adjusted closed counts,

P_{a d j}

= adjusted paid

On-level premium factor from cumulative rate levels

OLF_{y} = \frac{R L _{c u r}}{R L _{y}}, OLEP_{y} = E P_{y} \times OLF_{y}

, RL = cumulative rate-level index, cur = current year, y = accident year, EP = earned premium, OLEP = on-level EP

Disposal (claim settlement) rate at a given maturity

d = \frac{C}{U}

, d = disposal rate, C = closed claim counts at the maturity, U = ultimate (or reported) claim counts

Implied average unpaid per open-plus-IBNR claim

Avg Unpaid = \frac{Case O/S + IBNR}{Open Counts + Unreported Counts}

, Case O/S = case outstanding reserve, IBNR = incurred-but-not-reported reserve

Roll-forward reserve identity with prior-year development

R_{c u r r e n t} = R_{p r i or} - P_{p er i o d} + Δ U

, R = held reserve, P = paid during period, ΔU = new ultimate − prior ultimate (development)

Volume-weighted age-to-age development factor

f_{k} = \frac{\sum _{i} C _{i, k + 1}}{\sum _{i} C _{i, k}}

f_{k}

= factor from maturity k to k+1,

C_{i, k}

= losses for accident year i at maturity k

Cumulative development factor to ultimate including the tail

CDF_{ult} = (\prod_{k} f_{k}) \times f_{tail}

f_{k}

= observed age-to-age factor at age k,

f_{tail}

= tail factor from last observed age to ultimate

Surplus impact of a reserve error via reserve-to-surplus leverage

E_{s} = E_{r} \times \frac{R}{S}

E_{s}

= surplus error %,

E_{r}

= reserve error %, R = carried reserves, S = surplus

Age-to-age link factor on a cumulative triangle

f_{t \to t + 1} = \frac{C _{t + 1}}{C _{t}}

C_{t}

= cumulative losses at age t,

C_{t + 1}

= cumulative losses at next age; computed only on cumulative data

Average case outstanding diagnostic

Avg Case O/S = \frac{Reported - Paid}{Open Counts}

, Reported = cumulative reported losses, Paid = cumulative paid losses, Open Counts = open claim counts

Ceded loss on an excess-of-loss layer

L_{ce d e d} = min (max (L_{g r oss} - R, 0), ℓ)

L_{g r oss}

= gross loss, R = retention (attachment),

ℓ

= layer limit

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